What are Surfaces?

Differential Geometry · Axiom Academy

Discover the fundamental objects of differential geometry through interactive exploration. Rotate the sphere below to explore your first surface in 3D space. Click and drag on the sphere to rotate it. Notice how it looks smooth and continuous from every angle. Click on each surface to explore different examples. Each has unique properties! Adjust the zoom level to see how surfaces appear different at different scales. How Surfaces Differ from Curves Compare a 1-dimensional curve to a 2-dimensional surface. Parametrization: From Flat Maps to Curved Surfaces See how a flat rectangular region in the uv-plane maps to a curved surface in 3D space. A 2-dimensional object in 3D space that looks locally flat but can have global curvature and topology. Spheres, planes, tori, cylinders—each with unique geometric properties and curvature characteristics. Local properties (zoom in, study with calculus) vs global properties (step back, study with topology). Unlike 1D curves (one parameter), surfaces need two parameters to locate any point. Maps from flat (u,v) coordinates to 3D space, enabling us to apply calculus to curved surfaces.

This is the written version of the interactive lesson above. See the full Differential Geometry course.